Nevanlinna--Pick norms on scattered and Cantor spectra

Abstract

We introduce Nevanlinna--Pick norms associated with finite families of characters on a commutative semisimple Banach algebra $A$ and study the class $NP_\infty$ of algebras for which all these norms coincide with the $\ell^\infty$ norm. Our positive result is topological: if $A\in NP_\infty$ and $K\subset \Delta(A)$ is compact scattered Hausdorff, then the restriction algebra $NP(A,K)$ is isometrically isomorphic to $C(K)$. In particular, if $\Delta(A)$ is compact scattered, then \[ A\in NP_\infty \iff A\cong C(\Delta(A)) \quad\text{isometrically}. \] By contrast, we show that this rigidity fails in the presence of Cantor structure: for every compact Hausdorff space $S$ containing a Cantor subset, there exists a commutative semisimple unital Banach algebra $A\in NP_\infty$ with $\Delta(A)\cong S$ such that $A\neq C(S)$. As a consequence, for compact metrizable spectra, scatteredness is exactly the topological condition forcing rigidity.